![](/files/happy5.png)
Rainbow AP(4) in an almost equinumerous coloring
Problem Do 4-colorings of
, for
a large prime, always contain a rainbow
if each of the color classes is of size of either
or
?
![$ \mathbb{Z}_{p} $](/files/tex/ca03dcde1fc73a1d3d1916bca138cd11161ea69a.png)
![$ p $](/files/tex/928cd9d544fdea62f88a627aaee28c416c4366c0.png)
![$ AP(4) $](/files/tex/abd6fa4428454b30450d94292e000c9ecaa7a4fc.png)
![$ \lfloor p/4\rfloor $](/files/tex/d906d60e3b848dd93ec8be196b73063411f71d25.png)
![$ \lceil p/4\rceil $](/files/tex/5e7562752899a961fe9ccacfd0a84316504a88c2.png)
It is known that there are equinumerous colorings of (i.e. colorings of
for some
such that each color occurs
times) within which we cannot find rainbow arithmetic progressions of length
. ([CJR])
Bibliography
*[C] David Conlon, Rainbow solutions of linear equations over , Discrete Mathematics, 306 (2006) 2056 - 2063.
[CJR] David Conlon, Veselin Jungic, Rados Radoicic, On the existence of rainbow 4-term arithmetic progressions, Graphs and Combinatorics, 23 (2007), 249-254
* indicates original appearance(s) of problem.
Tight hypergraphs
It deservs to be mentioned that in any
-colouring of
, the equation
, have an heterochromatic (rainbow) solution; here,
denotes the multiplicative group of the field
.